| L(s) = 1 | + 4·4-s + 1.65i·7-s + 5.75i·13-s + 16·16-s + 33.4i·19-s − 25·25-s + 6.61i·28-s + 19.0·31-s + 57.1·37-s + 74.3i·43-s + 46.2·49-s + 23.0i·52-s + 96.5i·61-s + 64·64-s − 133.·67-s + ⋯ |
| L(s) = 1 | + 4-s + 0.236i·7-s + 0.442i·13-s + 16-s + 1.76i·19-s − 25-s + 0.236i·28-s + 0.614·31-s + 1.54·37-s + 1.72i·43-s + 0.944·49-s + 0.442i·52-s + 1.58i·61-s + 64-s − 1.99·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.366054949\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.366054949\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 4T^{2} \) |
| 5 | \( 1 + 25T^{2} \) |
| 7 | \( 1 - 1.65iT - 49T^{2} \) |
| 13 | \( 1 - 5.75iT - 169T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 - 33.4iT - 361T^{2} \) |
| 23 | \( 1 + 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 19.0T + 961T^{2} \) |
| 37 | \( 1 - 57.1T + 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 - 74.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 2.20e3T^{2} \) |
| 53 | \( 1 + 2.80e3T^{2} \) |
| 59 | \( 1 + 3.48e3T^{2} \) |
| 61 | \( 1 - 96.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 133.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 5.04e3T^{2} \) |
| 73 | \( 1 + 121. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 103. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 6.88e3T^{2} \) |
| 89 | \( 1 + 7.92e3T^{2} \) |
| 97 | \( 1 - 169T + 9.40e3T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.986683423336013727541110709207, −9.006887177600726230487403381454, −7.895606843921415146396324262433, −7.49356135218915929919081168608, −6.11414557186086794085527546452, −6.02089997327918589203840451224, −4.53226067562324775369873987508, −3.45974744652644831694851926564, −2.37681154817842057290673960728, −1.36260788810416995868711225711,
0.72411747146493428288439255704, 2.17400078564157633569288114034, 3.03745114118652260692204529217, 4.22019734708485391048983466363, 5.36819593177796458975530983818, 6.24435649779177956658788735316, 7.07777552426514790006492524417, 7.70993909221595968442232693052, 8.663500235629493092696761781060, 9.663773267965943455741653747727
